International Journal of Geometric Methods in Modern Physics

We study the Dirac and the chargeless Klein–Gordon–Fock equations in the geometry of Grumiller’s spacetime that describes a model for gravity of a central object at large distances. The Dirac equation is separated into radial and angular equations by adopting the Newman–Penrose formalism. The angular part of the both wave equations are analytically solved. For the radial equations, we managed to reduce them to one dimensional Schrödinger-type wave equations with their corresponding effective potentials. Fermions’s potentials are numerically analyzed by serving their some characteristic plots. We also compute the quasinormal frequencies of the chargeless and massive scalar waves. With the aid of those quasinormal frequencies, Bekenstein’s area conjecture is tested for the Grumiller black hole. Thus, the effects of the Rindler acceleration on the waves of fermions and scalars are thoroughly analyzed.

In this paper, we give $G$-invariant Einstein metrics on a class of homogeneous manifolds $G/K_1$, and then prove that every homogeneous manifold $G/K_1$ admits at least three families of $G$-invariant non-Riemannian Einstein–Randers metrics.

Stimulating from the hyperbolic octonionic generalization of the Maxwell-type equations of compressible fluids, an alternative reformulation has been proposed for the analogous multifluid plasma equations. In this paper, using both the fluid and electromagnetic behavior of the plasma, the compact and elegant expressions have been derived in terms of hyperbolic octonions as previously given for electromagnetic theory, linear gravity and fluid mechanics. Using the advantages of proposed model, the field equations of multifluid plasma have been combined in a single equation. Furthermore, the plasma wave equations in terms of generalized vorticity and Lamp vector have been expressed in a form similar to electromagnetic, gravitational counterparts previously given in relevant literature.

The main result of this paper is the proof that there are local electric and magnetic field configurations expressed in terms of field lines on an arbitrary hyperbolic manifold. This electromagnetic field is described by (dual) solutions of the Maxwell’s equations of the Einstein–Maxwell theory. These solutions have the following important properties: (i) they are general, in the sense that the knot solutions are particular cases of them and (ii) they reduce to the electromagnetic fields in the field line representation in the flat space-time. Also, we discuss briefly the real representation of these electromagnetic configurations and write down the corresponding Einstein equations.

In this paper, the Lie approximate symmetry analysis is applied to investigate the new exact solutions of the Rayleigh-wave equation. The power series method is employed to solve some of the obtained reduced ordinary differential equations with a small parameter. We yield the new analytical solutions with small parameter which is effectively obtained by the proposed method. The concept of nonlinear self-adjointness is used to construct the conservation laws for Rayleigh-wave equation. It is shown that this equation is approximately nonlinearly self-adjoint and therefore desired conservation laws can be found using appropriate formal Lagrangians.

We associate the lepton–quark families with the vertices of the 4D polytopes 5-cell ${(0001)}_{A4}$ and the rectified 5-cell ${(0100)}_{A4}$ derived from the $SU(5)$ Coxeter–Dynkin diagram. The off-diagonal gauge bosons are associated with the root polytope ${(1001)}_{A4}$ whose facets are tetrahedra and the triangular prisms. The edge-vertex relations are interpreted as the $SU(5)$ charge conservation. The Dynkin diagram symmetry of the $SU(5)$ diagram can be interpreted as a kind of particle-antiparticle symmetry. The Voronoi cell of the root lattice consists of the union of the polytopes ${{{{(1000)}_{A4}+(0100)}_{A4}+(0010)}_{A4}+(0001)}_{A4}$ whose facets are 20 rhombohedra. We construct the Delone (Delaunay) cells of the root lattice as the alternating 5-cell and the rectified 5-cell, a kind of dual to the Voronoi cell. The vertices of the Delone cells closest to the origin consist of the root vectors representing the gauge bosons. The faces of the rhombohedra project onto the Coxeter plane as thick and thin rhombs leading to Penrose-like tiling of the plane which can be used for the description of the 5-fold symmetric quasicrystallography. The model can be extended to $SO(10)$ and even to $SO(11)$ by noting the Coxeter–Dynkin diagram embedding $A_4\subset D_5\subset B_5$. Another embedding can be made through the relation $A_4\subset D_5\subset E_6$ for more popular $GU{T}^{\prime }s$. Appendix A includes the quaternionic representations of the Coxeter–Weyl groups $W(A_4)\subset {W(H}_4)$ which can be obtained directly from ${W(E}_8)$ by projection. This leads to relations of the $SU(5)$ polytopes with the quasicrystallography in 4D and $E_8$ polytopes. Appendix B discusses the branching of the polytopes in terms of the irreducible representations of the Coxeter–Weyl group $W(A_4)\approx S_5$.

The problem of a spineless charged particle with a time-dependent decaying mass interacting with a Coulomb and an inverse quadratic potentials is considered. The Green’s function is explicitly evaluated. The energy levels as well as the wave functions for the bound states are exactly determined.

We consider the distribution of spherically symmetric self-gravitating non-dissipative (but anisotropic) fluids under the expansion-free condition which requires the existence of vacuum cavity within the fluid distribution. The Darmois junction condition is investigated for matching the spherically symmetric metric to an internal vacuum cavity (Minkowski space-time). We have studied some analytical models, total of three family of solutions out of which two satisfy the junction conditions over both the hypersurfaces. The models are investigated under some known dynamical assumptions which further provide analytical solution in each family.

We study right-invariant (respectively, left-invariant) Poisson–Nijenhuis structures ($P$-$N$) on a Lie group $G$ and introduce their infinitesimal counterpart, the so-called r-n structures on the corresponding Lie algebra $𝔤$. We show that $r$-$n$ structures can be used to find compatible solutions of the classical Yang–Baxter equation (CYBE). Conversely, two compatible $r$-matrices from which one is invertible determine an $r$-$n$ structure. We classify, up to a natural equivalence, all $r$-matrices and all $r$-$n$ structures with invertible $r$ on four-dimensional symplectic real Lie algebras. The result is applied to show that a number of dynamical systems which can be constructed by $r$-matrices on a phase space whose symmetry group is Lie group a $G$, can be specifically determined.

Nearly every known pair of isospectral but nonisometric manifolds — with as most famous members isospectral bounded $ℝ$-planar domains which makes one “not hear the shape of a drum” [M. Kac, Can one hear the shape of a drum? Amer. Math. Monthly 73(4 part 2) (1966) 1–23] — arise from the (group theoretical) Gassmann–Sunada method. Moreover, all the known $ℝ$-planar examples (so counter examples to Kac’s question) are constructed through a famous specialization of this method, called transplantation. We first describe a number of very general classes of length equivalent manifolds, with as particular cases isospectral manifolds, in each of the constructions starting from a given example that arises itself from the Gassmann–Sunada method. The constructions include the examples arising from the transplantation technique (and thus in particular the known planar examples). To that end, we introduce four properties — called FF, MAX, PAIR and INV — inspired by natural physical properties (which rule out trivial constructions), that are satisfied for each of the known planar examples. Vice versa, we show that length equivalent manifolds with FF, MAX, PAIR and INV which arise from the Gassmann–Sunada method, must fall under one of our prior constructions, thus describing a precise classification of these objects. Due to the nature of our constructions and properties, a deep connection with finite simple groups occurs which seems, perhaps, rather surprising in the context of this paper. On the other hand, our properties define in some sense physically irreducible pairs of length equivalent manifolds — “atoms” of general pairs of length equivalent manifolds, in that such a general pair of manifolds is patched up out of irreducible pairs — and that is precisely what simple groups are for general groups.

In this paper, we will consider the birth and evolution of fields during formation of $N$-dimensional manifolds from joining point-like ones. We will show that at the beginning, only there are point-like manifolds which some strings are attached to them. By joining these manifolds, 1-dimensional manifolds are appeared and gravity, fermion, and gauge fields are emerged. By coupling these manifolds, higher dimensional manifolds are produced and higher orders of fermion, gauge fields and gravity are emerged. By decaying $N$-dimensional manifold, two child manifolds and a Chern–Simons one are born and anomaly is emerged. The Chern–Simons manifold connects two child manifolds and leads to the energy transmission from the bulk to manifolds and their expansion. We show that $F$-gravity can be emerged during the formation of $N$-dimensional manifold from point-like manifolds. This type of $F$-gravity includes both type of fermionic and bosonic gravity. $G$-fields and also $C$-fields which are produced by fermionic strings produce extra energy and change the gravity.

The relativistic Lagrangian in presence of potentials was formulated directly from the metric, with the classical Lagrangian shown embedded within it. Using it we formulated covariant equations of motion, a deformed Euler–Lagrange equation, and relativistic Hamiltonian mechanics. We also formulate a modified local Lorentz transformation, such that the metric at a point is invariant only under the transformation defined at that point, and derive the formulae for time-dilation, length contraction, and gravitational redshift. Then we compare our formulation under non-relativistic approximations to the conventional ad hoc formulation, and we briefly analyze the relativistic Liénard oscillator and the spacetime it implies.

After reviewing the $f(T,{𝒯})$ gravity, in which $T$ is the torsion scalar and ${𝒯}$ is the trace of the energy-momentum tensor, we refer to two cosmological models of this theory in agreement with observational data. Thereinafter, we consider a flat Friedmann–Robertson–Walker (FRW) universe filled by a pressureless source and look at the terms other than the Einstein terms in the corresponding Friedmann equations, as the dark energy (DE) candidate. In addition, some cosmological features of models, including equation of states and deceleration parameters, are addressed helping us in getting the accelerated expansion of the universe in quintessence era. Finally, we extract the scalar field as well as potential of quintessence, tachyon, K-essence and dilatonic fields for both $f(T,{𝒯})$ models. It is observed that the dynamics of scalar field as well as the scalar potential of these models indicate an accelerated expanding universe in these models.

In this work we investigate which Loop Quantum Cosmology (LQC)-corrected Gauss–Bonnet $F({𝒢})$ gravity can realize two singular cosmological scenarios, the intermediate inflation and the singular bounce scenarios. The intermediate inflation scenario has a Type III sudden singularity at $t=0$, while the singular bounce has a soft Type IV singularity. By using perturbative techniques, we find the holonomy-corrected $F({𝒢})$ gravities that generate at leading order the aforementioned cosmologies and we also argue that the effect of the holonomy corrections is minor to the power spectrum of the primordial curvature perturbations of the classical theory.

Using dyonic solutions in the type IIA superstring theory on Calabi–Yau (CY) manifolds, we reconsider the study of black objects and quantum information theory using string/string duality in six dimensions. Concretely, we relate four-qubits with a stringy quaternionic moduli space of type IIA compactification associated with a dyonic black solution formed by black holes (BHs) and black 2-branes (B2B) carrying eight electric charges and eight magnetic charges. This connection is made by associating the cohomology classes of the heterotic superstring on $T^4$ to four-qubit states. These states are interpreted in terms of such dyonic charges resulting from the quaternionic symmetric space $\frac{\mathrm{\text{SO}}(4,4)}{\mathrm{\text{SO}}(4)\times \mathrm{\text{SO}}(4)}$ corresponding to a $N=4$ sigma model superpotential in two dimensions. The superpotential is considered as a functional depending on four quaternionic fields mapped to a class of Clifford algebras denoted as ${\mathrm{\text{Cl}}}_{0,4}$. A link between such an algebra and the cohomology classes of $T^4$ in heterotic superstring theory is also given.

A new model which allows a non-minimal coupling between gravity and quintessence in the configuration of teleparallel gravity was recently proposed by Geng et al. [“Teleparallel” dark energy, Phys. Lett. B 704 (2011) 384–387] and they named it teleparallel dark energy. Now the main problem which arises is to know what is the source of this dark energy? The answer of this question is given by us in $M$-theory. This type of dark energy may be produced at three stages in our model. First, one six-dimensional universe is formed by combining and expanding $D0$-branes. We know that this universe-brane is polarized on two circles and our four-dimensional cosmos and two $D1$-branes are yielded. At third stage, two $D1$-branes glued to each other and one $D2$-brane is formed. This $D2$ connects our universe with another universe, gives its energy to them and causes the production of dark energy. Thus, the $D2$-brane is unstable and dissolves in our four-dimensional universes and supplies the needed teleparallel dark energy for expansion. These calculations are extended to $M$-theory and shown that the amount of teleparallel dark energy which is produced by compactification of universe-branes in $M$-theory is more than string theory.

We consider the recently proposed higher derivative torsion corrected modified teleparallel gravity and holographic dark energy (HDE) models. We apply the correspondence scheme to construct models in underlying scenario using various scale factor forms. We investigate the reconstructed functions through equation of state (EoS) parameter. It is demonstrated that the EoS parameter provides quintom-like nature of the Universe in most of the cases, i.e. it drives the Universe from vacuum dark energy era toward phantom era of the Universe by crossing the phantom divide line. We also demonstrate that the consistency with the observational data can be achieved.

We study the non-lightlike ruled surfaces in Minkowski 3-space with non-lightlike base curve $c(s)=\int (\alpha t+\beta n+\gamma b)ds$, where $t$, $n$, $b$ are the tangent, principal normal and binormal vectors of an arbitrary timelike curve $\mathrm{\Gamma }(s)$. Some important results of flat, minimal, II-minimal and II-flat non-lightlike ruled surfaces are studied. Finally, the following interesting theorem is proved: the only non-zero constant mean curvature (CMC) non-lightlike ruled surface is developable timelike ruled surface generated by binormal vector.